It’s impossible to confirm whether Arthur’s artificial estimate is actually in the right ballpark, because Amazon does not release sales numbers. But if the figure is close, it would also suggest that Amazon sells very few on-contract phones through its website, even those made by other manufacturers like Samsung.

Although Amazon does not release sales numbers, it does publish hourly updated lists of top-selling products in various categories. Take a look at Amazon’s bestselling contract cell phones: the 32 GB Fire Phone is in first place, as it has been since launch day. But flip over to the chart for best selling electronics in general, and you won’t find the Fire Phone — on contract or unlocked — anywhere on the top 100, although there are Amazon tablets represented there, as well as a few inexpensive unlocked smartphones.

If the Fire Phone tops Amazon’s bestselling contract cell phone list with an estimated maximum of 35,000 devices sold in a month, even with premium front page placement on Amazon.com, the number two device — the Galaxy S5 with Verizon service — probably isn’t doing gangbusters either.

Fire on the Amazon

Like retail giants Wal-Mart and Best Buy, Amazon is an authorized retailer for service plans and devices from AT&T, Verizon, and Sprint. It’s been in this space since 2009. There are a variety of reasons why consumers might not want their next device from Amazon: Return policies and fine print, as well as tasks like grandfathering your old contract or porting a number across carriers, become more complicated when you go through a third-party reseller. People might really prefer to upgrade their devices in person. And Amazon does not sell iPhones on contract.

But using Amazon’s wireless storefront is actually a pleasant experience — with common options clearly laid out — and has a decent selection across all carriers. Plus, it frequently runs deals that undercut the carriers. So it might be that online on-contract retailing is just not a great business: One big independent online reseller, LetsTalk, is no longer taking orders at all. As recently as three years ago Wirefly described itself as a “the Internet’s number one authorized retailer of cell phones and plans” but now it’s a comparison service that just links out to the carriers. Amazon used to have a standalone storefront called Amazon Wireless for its carrier contract products, complete with a link in the homepage footer next to Zappos and Woot, but on-contract phones have since been folded into the main website.

Why did Amazon make the Fire Phone expensive?

The apparent weakness of Amazon’s wireless business calls into question the overall strategy that brought the Fire Phone to market. The up-front cost of the Fire Phone — $650 — means that buying it on contract is the only realistic option for most people. It’s easy to assume that Amazon, since it is an e-commerce behemoth that sells everything, would sell a bunch of Fire Phones as well. But a high-end phone on contract is a more complicated purchase than a Kindle.

Before the Fire Phone was launched, some observers predicted that it would be sold unlocked or with some kind of innovative plan that would shake up the current postpaid subsidy paradigm. Instead, the Fire Phone is sold through the same subsidy system that predates the original iPhone, but with a heavy emphasis on online sales because, well, it’s Amazon.

Perhaps Amazon doesn’t put much weight on its on-contract cell phone sales or it’s planning to emphasize the unlocked Fire Phone after a price cut or an update. But when it launched the standalone Amazon Wireless storefront in 2009 it certainly seemed like a targeted growth area for the company.

It’s not crazy for a device maker or retailer to embrace the current American upgrade cycle and postpaid landscape — Apple has recently been looking to pump up retail iPhone sales in its physical and online stores as well. Perhaps one of the myriad conflicting goals behind the Fire Phone was to establish that it’s realistic to buy an on-contract phone online from Amazon. If so, it doesn’t seem to have worked so far.

This article has been updated to clarify that the Fire Phone is available from other retailers in addition to Amazon and AT&T.